4 JAGS Stuff

From the report introduction:

There are two advantages to our proposed system compared with the current workflow:

  1. Our system combines data from multiple sources into a statistical model that includes uncertainty using Bayesian statistics.
  2. The operator can interact with the internal model through Excel to conduct scenario analysis and automatically visualise the results.

— Logan

This Rmarkdown-generated page will serve as proof that a fully automated proof of concept has been developed. Whether the code is sufficiently commented or not … is a different question.

4.1 Setup

configpath = '../wairakei_data/config.xlsx'
regdatapath = '../wairakei_data/data.xlsx'
extraliqregpath = "../wairakei_data/extra_liq.csv" # for regression
extradatapath = "../wairakei_data/well_pi.csv"     # ts data
pipath <- "../wairakei_data/short version Generation Projection 2016.xlsx"

base_year = '2000'
prediction_date = '2017-12-01'
production_curve_wells = c('wk255', 'wk263')
tsplotwells = c("wk118", "wk216", "wk605")
decline_wells = c(production_curve_wells, "wk272", "wk86", "wk116")
base_datetime = as.POSIXct(paste(base_year, 1, 1, sep='-'))
today_datetime = as.POSIXct(prediction_date)
# theme_update(text=element_text(family="Times New Roman"))
'%ni%' <- Negate('%in%')
# for over-plotting
special_wells = c(production_curve_wells, tsplotwells, "wk86", "wk116")
use.censor = F

n_steps = 1000

censor = function(x, type) {
  # Hash the facility identifier (beware of hash clashes)
  if (!use.censor) {
    return(x)
  } else if (type=="well") {
    return(paste0("w", toupper(substr(sha1(x), 1, 3))))
  } else if (type=="fp") {
    return(paste0("fp", toupper(substr(sha1(x), 1, 2))))
  }
}

4.2 Data Handling

Data is extracted and cleaned using Python in simulation.ipynb. The Python notebook is also used to generate a rudimentary config file, but some things (network connectivity) are specified manually.

R is used to:

  • Read raw data and config from Excel/CSV files
  • Do additional pre-processing that depends on the data available
  • Censor sensitive facility names
  • Create a graph structure
  • Make the data into a JAGS-readable format

4.2.1 Load Data

Reads data from several spreadsheets, including PI data. PI data is special because it has not been pre-processed. It requires additional filtering and basic pre-processing.

# read in config data
configsheets = excel_sheets(configpath)
for (sheet in configsheets) {
  assign(sheet, read_excel(configpath, sheet))
}
stopifnot(!anyDuplicated(well_fp_map$well)) # each well cannot map to multiple flash plants

# read in PI data
PI <- read_excel(pipath, "From PI sheet", skip=1) %>%
  rename(facility = Unit,
         variable = X__1,
         id = X__2,
         description = X__3,
         code = X__4) %>%
  gather(key="datechar", value="value", -c(facility, variable, id, description, code)) %>%
  mutate(date = as.Date(as.numeric(datechar), origin = "1899-12-30"),
         value = as.numeric(value)) %>% select(-c(datechar, id)) %>%
  mutate_if(is.character, tolower) %>%
  mutate(value = as.numeric(value)) %>%
  drop_na(value) %>%
  filter(date >= as.Date("2017-11-01"), date < as.Date(prediction_date)) %>%
  filter(!str_detect(variable, "condition|calc")) %>%
  filter(str_detect(facility, "wk")) 
extra_liq <- PI %>%
  select(facility, date, variable, value) %>% 
  # filter(value>1e-4) %>%
  filter(str_detect(variable, "plot|phase|whp|flow")) %>%
  spread(key=variable, value=value) %>%
  mutate(mf = pmax(`2phase flow`, `fp14  plot flow`, `fp15  plot flow`, `flow`, na.rm=T),
         whp = pmax(`fp14  plot whp`, `fp15  plot whp`, `fp16  plot whp`, `whp`, na.rm=T),
         source = "PI Database") %>%
  select(well=facility, date, whp, mf, source) %>%
  drop_na()

# read in regression data (plus extra)
regression_df = read_excel(regdatapath) %>% mutate(source="Well Tests")
dry_df = PI %>%
  filter(str_detect(facility, "wk")) %>%
  select(facility, date, variable, value) %>%
  # filter(value>1e-2) %>%
  group_by(facility, date) %>%
  spread(key=variable, value=value) %>%
  select(facility, date, `ip sf`, `actual massflow`) %>%
  gather(key="key", value="mf", `ip sf`, `actual massflow`) %>%
  ungroup() %>%
  drop_na() %>%
  rename(well=facility)

4.2.2 Censor names

Censor well and flash plant names using a hash algorithm. Change the flag in setup to disable.

dry_df$well = censor(dry_df$well, "well")
extra_liq$well = censor(extra_liq$well, "well")
fp_constants$fp = censor(fp_constants$fp, "fp")
fp_gen_map$fp = censor(fp_gen_map$fp, "fp")
operating_conditions$well = censor(operating_conditions$well, "well")
regression_df$well = censor(regression_df$well, "well")
well_fp_map$well = censor(well_fp_map$well, "well")
well_fp_map$fp = censor(well_fp_map$fp, "fp")

production_curve_wells = censor(production_curve_wells, "well")
special_wells = censor(special_wells, "well")
tsplotwells = censor(tsplotwells, "well")

4.2.3 Preprocessing

Generate metadata, such as which wells have which data sources, and translate facility names into unique integer IDs. Also creates dummy facilities for multiple purposes.

# combine with extra
regression_df = plyr::rbind.fill(regression_df, extra_liq)

regression_df = regression_df %>%
  mutate(date_numeric = as.numeric(date - base_datetime)) %>%
  mutate(date_numeric=ifelse(date_numeric>0, date_numeric, NA))  # remove dates before baseline
dry_df = dry_df %>%
  filter(well %ni% unique(regression_df$well)) %>%
  mutate(date_numeric = as.numeric(as.POSIXct(date) - base_datetime)) %>%
  mutate(date_numeric=ifelse(date_numeric>0, date_numeric, NA))  # remove dates before baseline
well_fp_map = well_fp_map %>% select(well, fp) %>% drop_na()

# today_numeric = (Sys.time() - base_datetime) %>% as.numeric()
today_numeric = (today_datetime - base_datetime) %>% as.numeric()

# assign unique facility IDs
liq_wells = unique(regression_df$well) # aka production curve wells
dry_wells = unique(dry_df$well)        # aka time series wells
map_wells = unique(well_fp_map$well)   # any well mapped in config

well_names = unique(c(liq_wells, dry_wells))
fp_names = c(well_fp_map$fp, fp_gen_map$fp, fp_constants$fp) %>% unique()

fluid_types = c('ip', 'lp', 'w')
gen_names = gen_constants$gen %>% unique() %>% sort()
ip_gen_names = paste(gen_names, 'ip', sep='_')
lp_gen_names = paste(gen_names, 'lp', sep='_')
w_gen_names = paste(gen_names, 'w', sep='_')
dummy_gen_names = c(ip_gen_names, lp_gen_names, w_gen_names) %>% sort()
all_names = c('DUMMY', well_names, fp_names, dummy_gen_names, gen_names)
ids = 1:length(all_names)
names(ids) = all_names

# check data quality
no_data_wells = map_wells[!map_wells %in% c(liq_wells, dry_wells)]  # see which ones we're completely guessing for
no_map_wells = c(liq_wells, dry_wells)[!c(liq_wells, dry_wells) %in% map_wells]
missing = data.frame(Wells = c(paste(no_map_wells, collapse=", ")),
                     row.names = c("Data available but no FP"))
print(xtable(missing, type = "latex",
             caption=paste0("Potential data quality issues. ", names(ids)[71], " is known to be not connected, and ", names(ids)[31], " has an A/B pairing with ", names(ids)[32], "."),
             label="tab:quality"),
      file = "../_media/quality.tex")

# add names in data with IDs
regression_df = regression_df %>% mutate(well_id=ids[well])
dry_df = dry_df %>% mutate(well_id=ids[well])
operating_conditions = operating_conditions %>% mutate(well_id=ids[well]) %>% rename(whp_pred=whp)
fp_constants = fp_constants %>% mutate(fp_id=ids[fp])
gen_constants = gen_constants %>% mutate(gen_id=ids[gen]) %>% select(-gen)
well_fp_map = well_fp_map %>% mutate(well_id=ids[well], fp_id=ids[fp]) %>% select(-c(well, fp))
fp_gen_map = fp_gen_map %>% mutate(fp_id=ids[fp], gen_ip_id=ids[gen_ip], gen_lp_id=ids[gen_lp], gen_w_id=ids[gen_w]) %>% select(-c(fp, gen_ip, gen_lp, gen_w))

incomplete.fps = unique(well_fp_map %>%
  filter(is.na(well_id)) %>%
  mutate(fp = names(ids)[fp_id]) %>%
  pull(fp))

4.2.4 Graph

Work out which of the (now uniquely integer-identified) facilities flows to which. Then generates a graphic to check for correctness.

# create connectivity matrix. i flows to j
# wells to FPs
v = matrix(0, nrow=length(ids), ncol=length(ids))
v[1,-1] = 1
for (i in 1:nrow(well_fp_map)) {
  id_i = well_fp_map[[i, 1]]
  id_j = well_fp_map[[i, 2]]
  v[id_i, id_j] = 1
}
# send ip/lp/w flows to dummy gens
for (i in 1:nrow(fp_gen_map)) {
  id_i = fp_gen_map[[i, 1]]
  for (j in 2:ncol(fp_gen_map)) {
    facility_j = names(ids)[fp_gen_map[[i, j]]]
    facility_dummy_j = paste(facility_j, fluid_types[j-1], sep='_')
    id_j = ids[facility_dummy_j]
    if (!is.na(id_j)) {
      v[id_i, id_j] = 1
    }
  }
}
# dummy gens to gens
for (i in 1:nrow(gen_constants)) {
  id_j = gen_constants$gen_id[i]
  facility_j = names(ids)[id_j]
  for (fluid in fluid_types) {
    facility_dummy_i = paste(facility_j, fluid, sep='_')
    id_i = ids[facility_dummy_i]
    v[id_i, id_j] = 1
  }
}

# convert form
m = matrix(0, nrow=nrow(v), ncol=max(colSums(v)))
rownames(m) = all_names
for (i in 1:nrow(v)) {
  for (j in 1:ncol(v)) {
    if (v[[i, j]]==1) {
      m[j, sum(m[j,]>0)+1] = i
    }
  }
}
flows_to = function(well) {
  return(names(ids)[m[well,]][-1])
}

# generate coordinates
dummy_locs = data.frame(name='DUMMY', x=-0.1, y=0)
well_locs = data.frame(name=well_names, x=0, y=seq(1, 1/(length(well_names)-1), length.out=length(well_names)))
fp_locs = data.frame(name=fp_names, x=1, y=seq(0, 1, length.out=length(fp_names)))
gen_dummy_locs = data.frame(name=dummy_gen_names, x=2, y=seq(0, 1, length.out=length(dummy_gen_names)))
gen_locs = data.frame(name=gen_names, x=2.5, y=seq(1/11, 10/11, length.out=length(gen_names)))
locs = rbind(dummy_locs, well_locs, fp_locs, gen_dummy_locs, gen_locs)
locs$id = ids[locs$name]
locs = locs %>% arrange(id)

g = graph_from_adjacency_matrix(v) %>%
  set_vertex_attr('label', value=all_names) %>%
  set_vertex_attr('x', value=as.vector(locs$x)) %>%
  set_vertex_attr('y', value=as.vector(locs$y)) %>%
  set_vertex_attr('label.degree', value=pi) %>%
  as.undirected()
V(g)$size = ifelse(V(g)$label %in% well_names, 4, 8)
V(g)$color = ifelse(V(g)$label %in% dry_wells, "red", ifelse(V(g)$label %in% no_data_wells, "grey", "orange"))
E(g)$color = "black"
E(g)[which(tail_of(g, E(g))$label=="DUMMY")]$color = "grey"

# png("../_media/full_network.png")
# par(mar=c(0,3,0,0), family="Times")
# plot(g, vertex.label.dist=3,
#      mark.groups = list(wells=ids[well_names], fps=ids[fp_names], gens=ids[gen_names]),
#      mark.col = "#DDDDDD",
#      mark.border = NA)
# text(c(-1, -0.3, 0.4, 0.9), 1.15, c("Wells", "Flash plants", "Dummy gens", "Generators"), cex=1.25)
# dev.off()
plot(g, vertex.label.dist=3,
     mark.groups = list(wells=ids[well_names], fps=ids[fp_names], gens=ids[gen_names]),
     mark.col = "#DDDDDD",
     mark.border = NA)

The dummy node is necessary because when indexing a subset of flows that go into a node, this subset cannot be empty. The dummy node has zero mass flowing out of it.

4.2.5 Format Data

JAGS requires data to be real numbers, vectors or matrices in a named list. It can also impute NA values from a distribution. Data wrangling is a significant part of the work - potentially more than the actual model coding and the results analysis combined.

This code also centers some of the covariates so it does not have to be done in JAGS.

\[\begin{equation} x_\text{whp} \leftarrow x_\text{whp} - \overline{x_\text{whp}} \end{equation}\]
regression_list = regression_df %>% select(well_id, whp, mf, date_numeric) %>% as.list()
dry_list = dry_df %>%
  filter(date < prediction_date) %>%
  rename(well_id_dry=well_id, mf_dry=mf, date_numeric_dry=date_numeric) %>% # use these in a different regression
  select(well_id_dry, mf_dry, date_numeric_dry) %>% as.list()
operating_conditions_list = operating_conditions %>% arrange(well_id) %>% select(whp_pred) %>% as.list()
fp_constants_list = as.list(fp_constants)
gen_constants_list = as.list(gen_constants %>% select(gen_id, factor))
facilities = data.frame(id=ids) %>%
  left_join(operating_conditions %>% rename(id=well_id) %>% filter(id %in% ids) %>% select(-well), by='id') %>%
  left_join(gen_constants %>% select(factor, id=gen_id), by='id') %>%
  left_join(fp_constants %>% rename(id=fp_id), by='id') %>%
  filter(id %in% ids) %>%  # in case extras specified in data
  mutate(mf_pred=NA) %>%
  mutate(n_inflows=colSums(v))

well_ids = ids[well_names]
liq_well_ids = ids[liq_wells]
dry_well_ids = ids[dry_wells]
fp_ids = ids[fp_names]
ip_gen_ids = ids[ip_gen_names]
lp_gen_ids = ids[lp_gen_names]
w_gen_ids = ids[w_gen_names]
gen_ids = ids[gen_names]

# force all mass to IP steam
dry_fps = c("poi dry", "direct ip")
dry_fp_ids = ids[dry_fps]
facilities$hf_ip[facilities$id %in% dry_fp_ids] = 10
facilities$hfg_ip[facilities$id %in% dry_fp_ids] = 10
facilities_list = facilities %>% select(-id) %>% as.list()

# experimental TS data matrix for dry wells
ar_order = 1
empty = setNames(data.frame(matrix(ncol = length(all_names), nrow = 0)), all_names)
drymatrix = dry_df %>% 
  select(well, date_numeric, mf) %>% 
  spread(well, mf) %>% 
  select(-date_numeric)
drymatrix = empty %>%
  full_join(drymatrix) %>%
  as.matrix()
ar_well_ids = which(complete.cases(t(drymatrix[1:(ar_order+1),])))
ar_wells = names(ids)[ar_well_ids]
# which wells can we not use AR for
dry_no_ar_wells = dry_wells[!dry_well_ids %in% ar_well_ids]
dry_no_ar_well_ids = ids[dry_no_ar_wells]

# insert production curve predictions
stopifnot(all(tsplotwells %in% dry_df$well))
tsplotwells = ar_wells
days_since_last = as.integer(today_datetime - as.POSIXct(max(dry_df$date)))
prod = expand.grid(whp_prod=seq(6, 16, length.out=10),
                  well_id_prod=ids[production_curve_wells])
ts = expand.grid(date_numeric_ts=seq(min(dry_df$date_numeric), max(dry_df$date_numeric)+days_since_last, length.out=10),
                 well_id_ts=ids[tsplotwells])
prod_list = prod %>% as.list
ts_list = ts %>% as.list

# extend matrix for prediction
drymatrix = rbind(drymatrix, matrix(NA, nrow=days_since_last, ncol=ncol(drymatrix)))

# combine into one list
data = c(regression_list, dry_list, facilities_list, prod_list, ts_list,
         list(well_ids=well_ids, liq_well_ids=liq_well_ids, 
              dry_well_ids=dry_well_ids, dry_no_ar_well_ids=dry_no_ar_well_ids,
              fp_ids=fp_ids,
              gen_ids=gen_ids, ip_gen_ids=ip_gen_ids, lp_gen_ids=lp_gen_ids, w_gen_ids=w_gen_ids,
              today_numeric=today_numeric, m=m, dummy=1,
              ts=drymatrix, ts_ar=drymatrix, ts_ema=drymatrix, ar_well_ids=ar_well_ids))
# data$whp_pred[is.na(data$whp_pred)] <- mean(data$whp_pred, na.rm=T)

# center covariates
mean_whp <- mean(data$whp, na.rm=T)
mean_date_numeric <- mean(data$date_numeric, na.rm=T)

data$whp_c <- data$whp - mean_whp
data$whp_pred_c <- data$whp_pred - mean_whp
data$whp_prod_c <- data$whp_prod - mean_whp
data$date_numeric_c <- data$date_numeric - mean_date_numeric
data$today_numeric_c <- data$today_numeric - mean_date_numeric
data$date_numeric_dry_c <- data$date_numeric_dry - mean_date_numeric
data$date_numeric_ts <- data$date_numeric_ts - mean_date_numeric

pidataplot = ggplot(regression_df %>% filter(source=="PI Database"), aes(x=whp, y=mf, color=well)) +
  geom_point() +
  labs(title=paste("PI Regression Data from", min(extra_liq$date), "to", max(extra_liq$date)),
       x="Well-head pressure (bar)", 
       y="Mixed-phase mass flow (T/h)",
       color="Well") +
  guides(color=guide_legend(ncol=2)) +
  ggsave('../_media/pi_data.png', width=24.7, height=12, units='cm')
ggplotly(pidataplot)

4.3 Model

JAGS accepts a model in a text string. It uses an R-like syntax, but is a declarative language not sequential. We do basic manipulation of the output traces.

code = "
data {
  D <- dim(ts)
}
model {
  ##############################################
  # fit individual regressions to liquid wells #
  ##############################################
  for (i in 1:length(mf)) {
    mu[i] <- Intercept[well_id[i]] + beta_whp[well_id[i]] * whp_c[i] + beta_date[well_id[i]] * date_numeric_c[i]
    mf[i] ~ dnorm(mu[i], tau[well_id[i]])
    mf_fit[i] ~ dnorm(mu[i], tau[well_id[i]])
    # mf_fit[i] ~ dnorm(mu[i]*measurement_error_factor[i], tau[well_id[i]])
    # measurement_error_factor[i] ~ dunif(0.9, 1.1)
  }
  # fit regression to dry wells
  for (i in 1:length(mf_dry)) {
    mu_dry[i] <- Intercept[well_id_dry[i]] + beta_date[well_id_dry[i]] * date_numeric_dry_c[i]
    mf_dry[i] ~ dnorm(mu_dry[i], tau[well_id_dry[i]])
    mf_dry_fit[i] ~ dnorm(mu_dry[i], tau[well_id_dry[i]])
    # measurement_error_factor_dry[i] ~ dunif(0.9, 1.1)
  }
  for (j in dry_well_ids) {
    Intercept[j] ~ dnorm(0, 1e-12)
    beta_date[j] ~ dnorm(0, 1e-12)
    tau[j] ~ dgamma(1e-12, 1e-12)
  }
  # experimental AR1 model for dry wells
  for (j in ar_well_ids) {
    for (t in 2:D[1]) {
      mu_ar[t,j] <- c_ar[j] + theta_ar[j]*ts_ar[t-1,j]
      ts_ar[t,j] ~ dnorm(mu_ar[t,j], tau_ar[j]) T(0,)
    }
    theta_ar[j] ~ dnorm(0, 1e-12)
    c_ar[j] ~ dnorm(0, 1e-12)
    tau_ar[j] ~ dgamma(1e-12, 1e-12)
  }
  # experimental EWMA model (use at your own risk)
  for (j in ar_well_ids) {
    for (t in 2:D[1]) {
      mu_ema[t,j] <- alpha*mu_ema[t-1,j] + (1-alpha)*ts_ema[t,j]
      ts_ema[t,j] ~ dnorm(mu_ema[t-1,j], tau_ema[j]) T(0,)
    }
    mu_ema[1,j] <- ts_ema[1,j]
    theta_ema[j] ~ dnorm(0, 1e-12)
    c_ema[j] ~ dnorm(0, 1e-12)
    tau_ema[j] ~ dgamma(1e-12, 1e-12)
  }
  alpha ~ dbeta(0.5, 0.5)

  # HIERARCHICAL
  # fills in for any missing wells
  for (j in liq_well_ids) {
    Intercept[j] ~ dnorm(mu_Intercept, tau_Intercept)
    beta_whp[j] ~ dnorm(mu_beta_whp, tau_beta_whp)
    # beta_whp2[j] ~ dnorm(mu_beta_whp2, tau_beta_whp2)
    beta_date[j] ~ dnorm(mu_beta_date, tau_beta_date)
    tau[j] ~ dgamma(1e-12, 1e-12)
    sd[j] <- 1/max(sqrt(tau[j]), 1e-12)
  }

  # fill in any missing data
  for (i in 1:length(mf)) {
    date_numeric_c[i] ~ dnorm(mu_date_numeric, tau_date_numeric)
  }
  mu_date_numeric ~ dnorm(0, 1e-12)
  tau_date_numeric ~ dnorm(1e-12, 1e-12)
  
  # set hyperparameters
  mu_Intercept ~ dnorm(0, 1e-12)
  mu_beta_whp ~ dnorm(0, 1e-12)
  # mu_beta_whp2 ~ dnorm(0, 1e-12)
  mu_beta_date ~ dnorm(0, 1e-12)
  tau_Intercept ~ dgamma(1e-12, 1e-12)
  tau_beta_whp ~ dgamma(1e-12, 1e-12)
  # tau_beta_whp2 ~ dgamma(1e-12, 1e-12)
  tau_beta_date ~ dgamma(1e-12, 1e-12)

  #####################################
  # production curve for verification #
  #####################################
  for (i in 1:length(whp_prod)) {
    mu_prod[i] <- Intercept[well_id_prod[i]] + beta_whp[well_id_prod[i]] * whp_prod_c[i] + beta_date[well_id_prod[i]] * today_numeric_c
    # mf_prod[i] ~ dnorm(mu_prod[i], tau[well_id_prod[i]])
    mf_prod[i] <- mu_prod[i]
  }
  for (i in 1:length(date_numeric_ts)) {
    mu_ts[i] <- Intercept[well_id_ts[i]] + beta_date[well_id_ts[i]] * date_numeric_ts[i]
    mf_ts[i] ~ dnorm(mu_ts[i], tau[well_id_ts[i]])
  }

  #########################################################
  # simple model to fill in missing FP enthalpy constants #
  #########################################################
  for (i in fp_ids) {
    # missing fp constants
    hf_ip[i] ~ dgamma(param[1], param[7])
    hg_ip[i] ~ dgamma(param[2], param[8])
    hfg_ip[i] ~ dgamma(param[3], param[9])
    hf_lp[i] ~ dgamma(param[4], param[10])
    hg_lp[i] ~ dgamma(param[5], param[11])
    hfg_lp[i] ~ dgamma(param[6], param[12])
  }
  for (i in c(1, well_ids)) { 
    h[i] ~ dgamma(param[13], param[14])
    whp_pred_c[i] ~ dnorm(param[15], param[16])
  } # missing well constants
  for (i in 1:16) { param[i] ~ dgamma(1e-12, 1e-12) }               # uniform priors

  ########################################
  # make predictions (the stuff we want) #
  ########################################
  mf_pred[dummy] <- 0  # dummy well
  ip_sf[dummy] <- 0
  lp_sf[dummy] <- 0
  wf[dummy] <- 0
  
  # use production curve
  for (j in liq_well_ids) {
    mf_pred[j] <- max(Intercept[j] + beta_whp[j] * whp_pred_c[j] + beta_date[j] * today_numeric_c, 0)
  }
  # use naive TS reg
  for (j in dry_well_ids) { #dry_no_ar_well_ids) {
    mf_pred[j] <- max(Intercept[j] + beta_date[j] * today_numeric_c, 0)
  }
  # use AR(1)
  # for (j in ar_well_ids) {
  #   mf_pred[j] <- mu_ar[D[1], j]
  # }

  for (i in fp_ids) {
    mf_pred[i] <- sum(mf_pred[m[i,1:n_inflows[i]]])
    h[i] <- sum(mf_pred[m[i, 1:n_inflows[i]]] * h[m[i, 1:n_inflows[i]]]) / ifelse(mf_pred[i]!=0, mf_pred[i], 1)

    ip_sf[i] <- min(max((h[i] - hf_ip[i]), 0) / hfg_ip[i], 1) * mf_pred[i]
    lp_sf[i] <- min(max((min(hf_ip[i], h[i]) - hf_lp[i]), 0) / hfg_lp[i], 1) * (mf_pred[i] - ip_sf[i])

    total_sf[i] <- ip_sf[i] + lp_sf[i]
    wf[i] <- mf_pred[i] - total_sf[i]
  }
  # dummy gens and actual gens
  for (i in ip_gen_ids) { mf_pred[i] <- sum(ip_sf[m[i, 1:n_inflows[i]]]) }
  for (i in lp_gen_ids) { mf_pred[i] <- sum(lp_sf[m[i, 1:n_inflows[i]]]) }
  for (i in w_gen_ids) { mf_pred[i] <- sum(wf[m[i, 1:n_inflows[i]]]) }
  for (i in gen_ids) {
    mf_pred[i] <- sum(mf_pred[m[i,1:n_inflows[i]]])
    power[i] <- mf_pred[i] / mu_factor[i]
    mu_factor[i] ~ dunif(0.95*factor[i], 1.05*factor[i])  # uncertainty from email
  }
  total_power <- sum(power[gen_ids])
}
"
# cat(code, file="model.txt")

vars =  c('mf_fit',
          'mf_dry_fit',
          'mf_ts',
          'mf_prod',
          'mf_pred',
          'beta_date',
          'sd',
          'power',
          'total_sf',
          'mu_ar',
          'ts_ar',
          'mu_ema',
          'ts_ema',
          'alpha',
          'ip_sf',
          'lp_sf',
          'wf',
          paste0('h[', fp_ids, ']'),
          paste0('mu_', c('Intercept', 'beta_whp', 'beta_date')),
          'total_power')
n_chains = 2
burn_in = 100

model = jags.model(textConnection(code), data, n.chains=n_chains)
## Compiling data graph
##    Resolving undeclared variables
##    Allocating nodes
##    Initializing
##    Reading data back into data table
## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 6259
##    Unobserved stochastic nodes: 3916
##    Total graph size: 29435
## 
## Initializing model
update(model, burn_in)
out = coda.samples(model, n.iter=round(n_steps/n_chains), variable.names=vars)
outmatrix = as.matrix(out)
outframe = as.data.frame(outmatrix) %>%
  gather(key=facility, value=value) %>%
  mutate(variable=gsub("\\[.*$", "", facility), facility=parse_number(facility, na=c("NA")))
outframe$facility = factor(names(ids)[outframe$facility])

4.4 Convergence Tests

One of the difficulties with MCMC approximations is they often require a burn-in (warm-up) period before settling into the stationary distribution of the Markov chain. Only the stationary distribution corresponds to the joint distribution we wish to sample from. In most practical uses, there is no way to predict convergence, so we diagnose convergence by monitoring the sample trace and running diagnostic tests.

4.4.1 Trace plots

Poor convergence or mixing is indicated by a strong trend at the beginning of the trace plot.

trace1 <- outframe %>%
  filter(variable=='mf_pred', facility==censor('wk256', "well")) %>%
  mutate(index = 1:nrow(.))
trace2 <- outframe %>%
  filter(variable=='total_power') %>%
  mutate(index = 1:nrow(.))
trace3 <- outframe %>%
  filter(variable=='mu_Intercept') %>%
  mutate(index = 1:nrow(.))
traceplot = ggplot(trace1, aes(x=index, y=value, color=variable)) +
  geom_line(alpha=0.75) +
  geom_line(alpha=0.75) +
  geom_line(alpha=0.75) +
  coord_cartesian(xlim = c(max(trace1$index)-1000, max(trace1$index))) +
  labs(title="Trace Plot (Single chain)", x="Iteration", y="Parameter value", color="Variable")# +
  # ggsave('../_media/trace_plot.png', width=24.7, height=8, units='cm')
ggplotly(traceplot)

4.4.2 Geweke

Geweke’s convergence diagnostic for MCMC samples tests for equality of the means in the first 10% and last 50% of the trace. The means will be equal if the sample is drawn from a stationary distribution, indicating the burn-in period has been successfully excluded.

If true univariate convergence has been achieved, we expect 95% of variables to pass Geweke’s test with a z-score less than 1.96 with 95% confidence.

random_var_ix = sample.int(ncol(outmatrix), 100) # 100 random var because it takes too long
geweke.out = geweke.diag(out[,random_var_ix])
geweke.df = data.frame(Index = 1:length(unlist(geweke.out)),
                       z = unlist(geweke.out[1])) %>%
  mutate(out = ifelse(abs(z)>1.96, T, F)) %>%
  drop_na()
proportion_out = sum(geweke.df$out) / nrow(geweke.df)
gewekeplot = ggplot(geweke.df, aes(x=Index, y=z)) +
  geom_point() +
  geom_hline(data=data.frame(value=c(1.96,-1.96)), aes(yintercept=value), color='red') +
  labs(title=paste0("Geweke z-score. ", round(proportion_out, 2)*100, "% of points lie outside the 95% confidence interval."))# +
  # ggsave('../_media/geweke.png', width=24.7, height=6, units='cm')
ggplotly(gewekeplot)

4.4.3 Gelman

The Gelman-Rubin convergence diagnostic gives the potential scale reduction factor (PSRF) for each parameter. This requires at least two independent chains and tests whether the chains have converged to identical distributions. If the chains have not converged, the scale reduction factors will have upper confidence limits greater than one. It is possible that when run indefinitely, the variance of the parameter estimate could shrink by the PSRF.

gelman.out = gelman.diag(out[,c(paste0('mf_pred[', 8:9, ']'), 'beta_date[9]', 'mu_beta_whp', 'mu_beta_date', 'mu_Intercept', 'total_power')])[[1]] %>% as.data.frame()
kable(gelman.out) %>% kable_styling()
Point est. Upper C.I.
mf_pred[8] 0.999511 1.001557
mf_pred[9] 1.000125 1.002242
beta_date[9] 1.016304 1.016913
mu_beta_whp 1.005811 1.031651
mu_beta_date 1.005967 1.023880
mu_Intercept 1.000574 1.005091
total_power 1.061181 1.189868

Some of the upper CIs are slightly greater than one, but not significantly. Large PSRFs are acceptable if they are in components of the network that do not affect parameters of interest.

4.4.4 Raftery

Raftery’s diagnostic gives the number of samples required to estimate a quantile (or credible interval) to a certain accuracy. In this notebook we only run 1000 samples so it says we do not have enough.

raftery.out = raftery.diag(out[,c(paste0('mf_pred[', 8:9, ']'), 'beta_date[9]', 'mu_beta_whp', 'mu_beta_date', 'mu_Intercept', 'total_power')])
raftery.out[[1]]
## 
## Quantile (q) = 0.025
## Accuracy (r) = +/- 0.005
## Probability (s) = 0.95 
## 
## You need a sample size of at least 3746 with these values of q, r and s

4.5 Posteriors

We generate density plots in their most basic forms without post-processing.

4.5.1 Well Mass Flow

g1 = ggplot(outframe %>% 
              filter(facility %in% well_names, variable=="mf_pred", value>0) %>%
              mutate(source = ifelse(facility %in% dry_wells, "PI time series", "Production curve")), 
            aes(x=value, fill=facility)) +
  geom_density(aes(y=..scaled..), alpha=0.5, color=NA) + xlim(0, NA) +
  facet_grid(source~.) +
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(title=paste("Posterior Well Mass Flows for", prediction_date), x="Mass flow (T/h)", y="Scaled density", fill="Facility")# +
  # ggsave('../_media/mf_wells.png', width=24.7, height=8, units='cm')
ggplotly(g1, tooltip=c('facility', 'value'))

4.5.2 Decline Rate

g2 = ggplot(outframe %>% filter(variable=="beta_date", facility %in% special_wells), aes(x=value, fill=facility)) +
  geom_density(alpha=0.5, color=NA) +labs(title="Posterior Decline Rate of Test Data", x="beta_date (T/h/Bar)", y="Density", fill="Facility") +
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank())# +
  # ggsave('../_media/beta_date.png', width=24.7, height=6, units='cm')
ggplotly(g2, tooltip=c('facility', 'value'))

4.5.3 FP Mass Flow

g4 = ggplot(outframe %>% filter(facility %in% gen_names, variable=="mf_pred", value>0), aes(x=value, fill=facility)) +
  geom_density(aes(y=..scaled..), alpha=0.5, color=NA) + xlim(0, NA) + 
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(title=paste("Posterior Generator Values for", prediction_date), x="Mass flow (T/h)", y="Scaled density", fill="Facility")# +
  # ggsave('../_media/mf_gens.png', width=24.7, height=10, units='cm')
ggplotly(g4, tooltip=c('facility', 'value'))

4.5.4 Gen Mass Flow

g5.actual = data.frame(facility = c("WRK", "THI", "POI", "BIN"),
                       value = c(121.73567, 172.18096, 51.53028, 9.98687))
g5 = ggplot(outframe %>% filter(facility %in% gen_names, variable=="power", value>0), aes(x=value, fill=facility)) +
  geom_density(aes(y=..scaled..), alpha=0.5, color=NA) + xlim(0, NA) +
  geom_vline(data=g5.actual, aes(xintercept=value, color=facility)) +
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(x="Power (MW)", y="Scaled density", fill="Facility")# +
  # ggsave('../_media/power_gens.png', width=24.7, height=10, units='cm')

ggplotly(g5, tooltip=c('facility', 'value'))
# tsgrob4.5 = grid_arrange_shared_legend(g4, g5, nrow=2, ncol=1, position = "right")
# ggsave('../_media/gens.png', tsgrob4.5, width=24.7, height=6, units='cm')

4.5.5 Gen Power

tb6 <- outframe %>% filter(variable=="sd") %>% select(facility, value) %>%
  mutate(well=factor(facility)) %>%
  group_by(well) %>%
  summarise(Mean = mean(value), 
            `Lower 2.5%` = quantile(value, 0.025), 
            `Upper 97.5%` = quantile(value, 0.975)) %>%
  mutate_if(is.numeric, round, 3) %>%
  inner_join(regression_df %>% mutate(well=factor(names(ids)[well_id])) %>% group_by(well) %>% summarise(n=n()), by="well")
g6 = ggplot(outframe %>% filter(variable=="sd") %>% filter(facility %in% special_wells), aes(x=value, fill=facility)) +
  geom_density(alpha=0.5, color=NA) + coord_cartesian(xlim=c(0, max(tb6$`Upper 97.5%`))) +
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(title="Posterior Flow Deviation Estimates", x="Standard deviation", y="Density", fill="Facility")# +
  # ggsave('../_media/standard_deviation.png', width=24.7, height=10, units='cm')
ggplotly(g6, tooltip=c('facility', 'value'))

4.6 Advanced Analysis

4.6.1 High Variance wells

nrow.source = function(df, facilityname, sourcename) {
  stopifnot(length(sourcename)==1)
  return(nrow(df %>% filter(well==facilityname, source==sourcename)))
}
well_summaries = outframe %>%
  filter(facility %in% well_names, variable=="mf_pred") %>%
  group_by(facility) %>%
  summarise(mean = mean(value),
            sd = sd(value),
            n_test = nrow.source(regression_df, unique(facility),"Well Tests"),
            n_pi = nrow.source(regression_df, unique(facility), "PI Database"),
            use.test = ifelse(n_test>0, "Test data", "No test data"),
            use.pi = ifelse(n_pi>0, "PI data", "No PI data")) %>%
  arrange(desc(sd))
well_summaries$production.curve = ifelse(well_summaries$facility %in% liq_wells, "Production curve", "Time series")

fp_summaries = list(fp14 = well_summaries %>% filter(facility %in% flows_to(censor('fp14', 'fp'))),
                    fp15 = well_summaries %>% filter(facility %in% flows_to(censor('fp15', 'fp'))),
                    fp16 = well_summaries %>% filter(facility %in% flows_to(censor('fp16', 'fp'))))
for (fp in names(fp_summaries)) {
  print(xtable(fp_summaries[[fp]] %>% select(-c(use.test, use.pi, production.curve)),
               type = "latex",
               caption=paste("Data methods feeding flash plant", censor(fp, 'fp')),
               label=paste0("tab:well_summaries_", fp)),
      table.placement = "H",
      file = paste0("../_media/summaries_", fp, ".tex"))
}

n_summaries = well_summaries %>%
  group_by(use.pi, use.test) %>%
  count()

sourceplot = ggplot(well_summaries, aes(x=1, y=log(sd))) +
  geom_boxplot(fill='steelblue') +
  geom_label(data=n_summaries, aes(x=-Inf, y=-Inf, hjust=0, vjust=0, label=paste0("n=", n), family="Times New Roman"), label.size=0, fill='white') +
  facet_grid(.~ use.pi + use.test) +
  theme(axis.text.x=element_blank(),
        axis.ticks.x=element_blank()) +
  labs(title="Differences in Production Error by Data Source", x="Production curve data source", y="log(standard deviation)")# +
  # ggsave('../_media/error_source.png', width=24.7*0.5, height=6, units='cm')
ggplotly(sourceplot)
sourcetab = well_summaries %>% select(facility, mean, sd, n_test, n_pi)
# print(xtable(sourcetab %>% head(), type = "latex",
#              caption="Upon inspection of the wells with the most variance, there is no immediate cause for high variance. This requires further investigation.",
#              label="tab:well_summaries"),
#       table.placement = "h",
#       file = "../_media/well_summaries.tex")
kable(cbind(sourcetab)) %>%
  kable_styling() %>%
  scroll_box(height = "400px")
facility mean sd n_test n_pi
wk86 81.5344741 61.3421227 4 0
wk28 88.7191863 49.1865621 26 0
wk242 321.9122120 46.6096937 28 0
wk65 26.7570149 42.1490342 2 0
wk27 163.8370066 37.6843482 36 0
wk81 60.1607344 27.4466959 24 0
wk116 76.0484841 26.6558492 10 0
wk26b 112.4015979 25.6578444 13 0
wk76 124.3262822 23.0842652 31 0
wk123 184.1871324 22.9574870 31 0
wk67 92.9119044 22.1894908 26 0
wk59 86.9360822 22.1636942 31 0
wk96 27.8387902 20.8344581 10 0
wk72 151.6238128 20.6750130 26 0
wk71 127.4461328 19.8326786 32 0
wk256 218.9385203 18.8662810 32 30
wk268 90.9012237 17.4071984 27 30
wk266 326.5706599 17.0555332 29 30
wk264 393.5393997 16.4000437 34 30
wk245 418.2219228 15.5064443 41 30
wk26a 119.5009793 14.5287864 16 0
wk265 335.9188636 14.4117511 34 30
wk267 308.6025505 14.0059520 31 30
wk255 377.3065824 13.9818617 41 30
wk83 133.4618281 13.4622589 26 0
wk74 156.6504648 12.7175994 26 0
wk269 133.2196999 11.0219560 25 30
wk235 206.0957866 10.3933763 20 0
wk70 151.4158667 10.0002375 45 0
wk55 65.1317321 9.9032863 47 0
wk262 633.1256048 9.5574724 36 30
wk244 196.7817051 9.4283939 37 30
wk229 201.3411656 9.1752904 92 0
wk124 252.5206841 8.9259854 31 0
wk222 105.9781842 8.4414258 44 0
wk243 375.1448829 8.2169925 52 30
wk247 232.9289526 6.6020166 48 30
wk271 86.5929859 6.5142536 6 30
wk207 150.3506614 6.2642318 18 0
wk250 25.9639557 5.9004806 0 30
wk239 132.8546240 5.8128930 18 0
wk46 43.9515350 5.2028818 18 0
wk260 295.9918124 5.0412982 33 30
wk261 250.3107907 4.6730155 37 30
wk253 269.9234974 4.5727659 32 30
wk270 460.2166869 4.0465622 5 30
wk258 198.6058922 3.2505974 38 30
wk263 221.6051927 3.2470226 34 30
wk259 309.0674371 3.1877495 37 30
wk272 208.3516971 3.0291599 7 30
wk254 226.6098609 2.3835320 39 30
wk605 34.4044858 1.3023662 0 0
wk606 22.5021979 1.0342556 0 0
wk237 14.3441589 0.9160045 0 30
wk233 13.6224547 0.5723686 0 30
wk241 45.4437549 0.4758516 0 30
wk249 1.1520835 0.3806202 0 0
wk238 60.2461360 0.3380969 0 30
wk610 6.6595603 0.2506634 0 0
wk607 1.4406459 0.2482484 0 0
wk234 30.9725253 0.2376167 0 30
wk25 6.6234142 0.1906251 0 0
wk251 22.6509078 0.1783918 0 30
wk240 23.5145992 0.1445570 0 30
wk228 14.9865455 0.1435981 0 30
wk236 26.3738049 0.0825879 0 0
wk216 8.1298059 0.0760767 0 0
wk232 0.0847269 0.0734232 0 30
wk252 58.8881359 0.0566079 0 30
wk118 9.5054893 0.0345031 0 0
wk604 0.7825628 0.0285853 0 0
wk92 0.0023263 0.0023603 0 0
wk101 0.0017559 0.0013574 0 0
wk66 0.0000000 0.0000000 0 0
wk88 0.0000000 0.0000000 0 0

4.6.2 Regression Fits

prod = as.data.frame(outmatrix) %>%
  select(contains('prod')) %>%
  gather(key=facility, value=value) %>%
  mutate(which=parse_number(facility)) %>%
  mutate(whp=data$whp_prod[which],
         well = names(ids)[data$well_id_prod[which]]) %>%
  rename(mf=value) %>%
  group_by(well, whp) %>%
  summarise(lower=quantile(mf, 0.025),
            upper=quantile(mf, 0.975),
            mean=mean(mf))

plotdata = regression_df %>%
  filter(well_id %in% ids[production_curve_wells]) %>%
  mutate(datetime = factor(as.Date(date))) %>%
  mutate(source = factor(source, levels=c("Well Tests", "PI Database")))

# regression plot
regplot = ggplot(prod, aes(x=whp)) +
  geom_line(aes(y=mean, color=well)) +
  geom_ribbon(aes(ymin=lower, ymax=upper, fill=well), alpha=0.25) +
  geom_point(data=plotdata, aes(y=mf, color=well, size=date, shape=source), alpha=0.5) +
  labs(title="Linear Regression on Test and PI Data", x="Well-head pressure (bar)", y="Mass flow (T/h)", color="Well", shape="Data source", size="Date", fill="Well") +
  coord_cartesian(xlim=c(min(plotdata$whp)*0.9,max(plotdata$whp)*1.1), ylim=c(0,max(plotdata$mf)*1.1))# +
  # ggsave('../_media/production_curve.png', width=24.7*0.48, height=24.7*0.48, units='cm')
ggplotly(regplot)

4.6.3 Time Series Plots

tsplotwells = ar_wells
ts_fit = as.data.frame(outmatrix) %>%
  select(contains('mf_ts')) %>%
  gather() %>%
  mutate(index = parse_number(key)) %>% select(-key) %>%
  group_by(index) %>%
  summarise(lower=quantile(value, 0.025),
            upper=quantile(value, 0.975),
            mean=mean(value)) %>%
  cbind(ts) %>%
  mutate(well = factor(names(ids[well_id_ts])),
         date_numeric = date_numeric_ts)

# actual observations
tsplotdata = dry_df %>%
  filter(well_id %in% ids[tsplotwells]) %>%
  mutate(datetime = factor(as.Date(date)),
         facility = well)

# experimental AR1 time series
ar_fit = as.data.frame(outmatrix) %>%
  select(contains("mu_ar")) %>%
  gather() %>%
  mutate(date_numeric = as.numeric(str_extract(key, "(?<=\\[)(.*?)(?=,)")) + min(dry_df$date_numeric) - 1,
         facility = names(ids)[as.numeric(str_extract(key, "(?<=,)(.*?)(?=\\])"))]) %>%
  select(facility, date_numeric, value) %>%
  group_by(facility, date_numeric) %>%
  summarise(mean=mean(value),
            lower=quantile(value, 0.025),
            upper=quantile(value, 0.975)) %>%
  filter(facility %in% tsplotwells)

# experimental EMA time series
ewma_fit = as.data.frame(outmatrix) %>%
  select(contains("mu_ema")) %>%
  gather() %>%
  mutate(date_numeric = as.numeric(str_extract(key, "(?<=\\[)(.*?)(?=,)")) + min(dry_df$date_numeric) - 1,
         facility = names(ids)[as.numeric(str_extract(key, "(?<=,)(.*?)(?=\\])"))]) %>%
  select(facility, date_numeric, value) %>%
  group_by(facility, date_numeric) %>%
  summarise(mean=mean(value),
            lower=quantile(value, 0.025),
            upper=quantile(value, 0.975)) %>%
  filter(facility %in% tsplotwells)

# find plot limits
tsmax = max(c(ts_fit$upper, ar_fit$upper, ewma_fit$upper))

lintsplot = ggplot(ts_fit, aes(x=date_numeric, color=well, fill=well)) +
  geom_line(aes(y=mean), linetype="dashed") +
  geom_ribbon(aes(ymin=lower, ymax=upper), color=NA, alpha=0.25) +
  geom_line(data=tsplotdata, aes(y=mf)) +
  geom_vline(aes(xintercept = max(tsplotdata$date_numeric)), linetype="dashed", color="red") +
  coord_cartesian(ylim=c(0, 60)) +
  labs(title=paste("Linear Time Series Regression for Selected Wells in PI"), x="Days since baseline (2000)", linetype="")# +
  # ggsave('../_media/dry_time_series.png', width=24.7, height=8, units='cm')

arplot = ggplot(ar_fit %>% filter(facility %in% tsplotwells), aes(x=date_numeric, y=mean, fill=facility, color=facility)) +
  geom_line(data=tsplotdata, aes(y=mf)) +
  geom_ribbon(aes(ymin=lower, ymax=upper), color=NA, alpha=0.5) +
  geom_line(linetype="dashed") + coord_cartesian(ylim=c(0, 60)) +
  geom_vline(aes(xintercept = max(tsplotdata$date_numeric)), linetype="dashed", color="red") +
  labs(title="AR(1) Experiment", x="Days since first date", y="Mass flow (T/h)") +
  ggsave('../_media/ar_experiment.png', width=24.7, height=8, units='cm')

ewmaplot = ggplot(ewma_fit, aes(x=date_numeric, y=mean, fill=facility, color=facility)) +
  geom_line(data=tsplotdata, aes(y=mf)) +
  geom_ribbon(aes(ymin=lower, ymax=upper), color=NA, alpha=0.5) +
  geom_line(linetype="dashed") + coord_cartesian(ylim=c(0, 60)) +
  geom_vline(aes(xintercept = max(tsplotdata$date_numeric)), linetype="dashed", color="red") +
  labs(title="EWMA Experiment", x="Days since first date")# +
  # ggsave('../_media/ewma_experiment.png', width=24.7, height=8, units='cm')

ggplotly(lintsplot)
ggplotly(arplot)
ggplotly(ewmaplot)
tsgrob = grid_arrange_shared_legend(lintsplot, arplot, ewmaplot, nrow=3, ncol=1, position = "bottom")

tsgrob
## TableGrob (2 x 1) "arrange": 2 grobs
##   z     cells    name              grob
## 1 1 (1-1,1-1) arrange   gtable[arrange]
## 2 2 (2-2,1-1) arrange gtable[guide-box]
# ggsave('../_media/ts_experiment.png', tsgrob, width=24.7, height=24, units='cm')

4.6.4 Goodness of fit (OLS regression)

liq_fit = as.data.frame(outmatrix) %>%
  select(contains('mf_fit')) %>%
  gather(key='index', value='fitted') %>%
  mutate(index=as.integer(parse_number(index))) %>%
  group_by(index) %>%
  summarise(lower=quantile(fitted, 0.025),
            upper=quantile(fitted, 0.975),
            Fitted=mean(fitted),
            std=sd(fitted)) %>%
  cbind(regression_df) %>%
  mutate(`Standardised residual` = (Fitted-mf)/std,
         Well = factor(names(ids[well_id])),
         Observed = mf) %>%
  gather(key="key", value="value", `Standardised residual`, Observed) %>%
  select(Well, key, Fitted, value, source)

diagplot = ggplot(liq_fit, aes(x=Fitted, y=value)) +
  geom_point(aes(color=Well, shape=Well)) + scale_shape_manual(values = rep_len(1:25, length(unique(liq_fit$Well)))) +
  geom_smooth(color='black') +
  facet_wrap(~key, scales="free") +
  geom_hline(data=data.frame(key="Standardised residual", value=c(1.96,-1.96)), aes(yintercept=value), color='red') +
  geom_abline(data=data.frame(key="Observed", a = 1, b = 0), aes(slope = a, intercept=b), color='red') +
  # coord_cartesian(ylim=c(-4, 4)) +
  labs(title="Diagnostic Plots", x="Fitted mass flow (T/h)", y="") +
  theme(legend.position = "bottom") +
  guides(color=guide_legend(nrow=3, byrow=T), shape=guide_legend(nrow=3, byrow=T))# +
  # ggsave('../_media/diagnostics.png', width=24.7, height=12, units='cm')
ggplotly(diagplot)
## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'
selectwells = liq_fit %>% group_by(Well, key) %>% summarise(fittedsd = sd(Fitted)) %>%
  arrange(desc(fittedsd)) %>% head(56*2) %>% pull(Well)

observedplot = ggplot(liq_fit %>% filter(key=="Observed", Well %in% selectwells), aes(x=Fitted, y=value)) +
  geom_point(aes(color=source), alpha=0.5) +
  geom_smooth(color=NA, alpha=0.5) +
  facet_wrap(~Well, scales="free") +
  # geom_hline(data=data.frame(key="Standardised residual", value=c(1.96,-1.96)), aes(yintercept=value), color='red') +
  geom_abline(data=data.frame(key="Observed", a = 1, b = 0), aes(slope = a, intercept=b)) +
  labs(title="Linear Regression Fit Plots Per Well", x="Fitted mass flow (T/h)", y="Observed mass flow (T/h)", color="Data source") +
  theme(legend.position = "bottom")# +
  # guides(color=guide_legend(nrow=3, byrow=T), shape=guide_legend(nrow=3, byrow=T)) +
  # ggsave('../_media/observed.png', width=24.7, height=24.7, units='cm')

stdresplot = ggplot(liq_fit %>% filter(key=="Standardised residual", Well %in% selectwells), aes(x=Fitted, y=value)) +
  geom_point(aes(color=source), alpha=0.5) +
  geom_smooth(color=NA, alpha=0.5) +
  facet_wrap(~Well, scales="free_x") +
  geom_hline(data=data.frame(key="Standardised residual", value=c(1.96,-1.96)), aes(yintercept=value), color='red') +
  # geom_abline(data=data.frame(key="Observed", a = 1, b = 0), aes(slope = a, intercept=b), color='red') +
  labs(title="Linear Regression Residual Plots Per Well", x="Fitted mass flow (T/h)", y="Standardised residual", color="Data source") +
  coord_cartesian(ylim=c(-5, 5)) + theme(legend.position="bottom")# +
  # guides(color=guide_legend(nrow=3, byrow=T), shape=guide_legend(nrow=3, byrow=T)) +
  # ggsave('../_media/stdres.png', width=24.7, height=24.7, units='cm')

ggplotly(observedplot)
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
ggplotly(stdresplot)
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
# stdres_min = liq_fit %>% filter(key=="Standardised residual") %>% pull(value) %>% min()
# stdres_max = liq_fit %>% filter(key=="Standardised residual") %>% pull(value) %>% max()
# ggplot(liq_fit %>% filter(key=="Standardised residual"), aes(x=value)) +
#   geom_density(fill="red", alpha=0.5, color=NA) +
#   geom_line(data=data.frame(x=seq(stdres_min, stdres_max, length.out=100)), aes(x=x, y=dnorm(x)))

4.6.5 Limits and Constraint Violations

sf.df <- outframe %>% 
  filter(str_detect(variable, "total_sf") & value > 0) %>% 
  droplevels()
limits = fp_constants %>%
  mutate(facility = names(ids)[fp_id]) %>%
  select(facility, limit) %>% 
  drop_na()

p.limits = sf.df %>%
  left_join(limits, by=c("facility")) %>%
  mutate(greater = value > limit) %>%
  group_by(facility) %>%
  summarise(p.greater = mean(greater)) %>%
  drop_na()

limitplot = ggplot(sf.df %>% filter(facility %ni% incomplete.fps), aes(x=value, fill=facility)) +
  facet_wrap(~facility, scales = "free_y", ncol=2) +
  geom_density(alpha=0.5, color=NA) +
  geom_vline(data=limits, aes(xintercept=limit), color="red") +
  geom_label(data=p.limits %>% filter(facility %ni% incomplete.fps), aes(x=-Inf, y=Inf, hjust=0, vjust=1, label=paste0("p(>lim)=", p.greater), family="Times New Roman"), color="black", label.size=0, fill='white') +
  theme(legend.position="none",
        axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(title="Posterior Flash Plant Mass Flows", x="Steam flow (T/h)", y="Density", fill="Flash plant", color="Steam flow limit")# +
  # ggsave('../_media/constraints.png', width=24.7, height=10, units='cm')
ggplotly(limitplot)

4.6.6 Flow Comparison

flow.df <- outframe %>% 
  filter(facility %in% fp_names) %>%
  filter(str_detect(variable, "mf_pred|ip_sf|lp_sf|wf") & value > 0) %>%
  mutate(variable=ifelse(variable=="mf_pred", "mf", variable),
         variable=factor(variable, levels=c("mf", "ip_sf", "lp_sf", "wf")))

comparison = fp_constants %>% select("fp", contains("verification")) %>%
  rename(facility=fp) %>%
  gather(key="variable", value="value", -facility) %>%
  mutate(variable = gsub("^verification_", "", variable),
         variable=factor(variable, levels=c("mf", "ip_sf", "lp_sf", "wf"))) %>%
  drop_na()

ps = flow.df %>%
  left_join(comparison, by=c("facility", "variable")) %>%
  mutate(greater = value.x > value.y) %>%
  group_by(facility, variable) %>%
  summarise(p.greater = mean(greater)) %>%
  mutate(variable=factor(variable, levels=c("mf", "ip_sf", "lp_sf", "wf"))) %>%
  drop_na()

verificationplot = ggplot(flow.df %>% filter(facility %ni% incomplete.fps), aes(x=value)) +
  geom_density(aes(y=..scaled.., fill=variable, color=variable), alpha=0.5, show.legend=F) +
  geom_vline(data=comparison %>% filter(facility %ni% incomplete.fps), aes(xintercept=value)) +
  geom_label(data=ps %>% filter(facility %ni% incomplete.fps), aes(x=-Inf, y=Inf, hjust=0, vjust=1, label=paste0("p(>x)=", p.greater), family="Times New Roman"), label.size=0) +
  facet_grid(facility~variable, scales="free", space="free_y") +
  theme(axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  labs(x="Value", y="Scaled density", title="Comparison Between Predicted FP Flows and Sample Data")# +
  # ggsave('../_media/verification.png', width=24.7, height=20, units='cm')
ggplotly(verificationplot)